Consider the system of equations
$$ 3a + 2b + c+ d =14 $$ $$ a^{2} + b^{2} + c^{2} + d^{2} =14.$$
Is there any way to find maximum value of $d$ using AM-GM inequality. I am not even able to think, how to start approaching this problem.
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$\begingroup$ Are $a,b,c,d \ge 0$? $\endgroup$– vvgCommented Nov 21, 2022 at 12:41
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$\begingroup$ No, these are any real numbers. $\endgroup$– Manoj KumarCommented Nov 21, 2022 at 12:43
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5$\begingroup$ Using Cauchy-Schwarz (or with AM-GM after expanding if you want): $(3a+2b+c)^2 \leq 14(a^2+b^2+c^2) \implies (14-d)^2 \leq 14(14-d^2)$. Can you finish it from here? $\endgroup$– LHFCommented Nov 21, 2022 at 12:56
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$\begingroup$ Can we say $d \le \frac{28}{15}$? $\endgroup$– Manoj KumarCommented Nov 21, 2022 at 13:07
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$\begingroup$ @Math__Nerd yes, that's the answer. $\endgroup$– LHFCommented Nov 21, 2022 at 13:09
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